Global existence and blow-up of Wk,p solutions for a class of nonlinear wave equations with dispersive term

In this paper, we study the initial boundary value problem utt-αuxxt-uxxtt=σ(ux)x,(α⩾0),x∈Ω,t>0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,u(0,t)=u(1,t)=0,t⩾0,where Ω=(0,1). We first prove that if σ(s)∈Ck(R)(k⩾2) and satisfies